There is no one solid answer to this question. You should check out the Ride skill for the details. Here is the "best" way to to fight from horseback at a basic level.
1. You need to have the ride skill.
2. You need a war trained mount.
3. You must make a DC 10 Ride check to have your war trained mount attack ; this is a free action for you and as well as some form of action for your mount. (See 5 below.)
4. You can guide your mount with your knees with a DC 5 Ride check to allow two hands for your actions.
5. Your mount can either Full Attack or Move and then Standard Attack.
6. You may take a move action and a standard action... or do a Full Attack after your mount moves to combat range.
7. Mounted Combat and other feats add additional options.
If you do not have a war trained mount, most of the above is not possible or it is much harder. The Handle Animal skill allows you to "push" untrained mounts in some instances for some of these actions.
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I think that the mounted combat rules state that the rider cannot attack more than once if the mount moves:
http://paizo.com/pathfinderRPG/prd/combat.html
"If your mount moves more than 5 feet, you can only make a single melee attack."
You cannot have your mount move and then make a full attack yourself which point 6 in the previous post may imply.
Aside from that, yes, both the rider and the mount have the usual set of actions each (move + standard or full-round).